On Completely Prime Left Ideals of Ore Extensions

摘要

The study of prime ideals has been an active area of research and a considerable work has been done in this direction during recent past. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized. We recall that a right ideal / of a ring R is called a prime right ideal if JK ? I for some right ideals J, K of R such that JI ? I, then either J ? I or K ? I. Also a right ideal P of R is said.to be completely prime right ideal if for any a,b e R such that aP ? P, ab ∈ P implies that a ∈ P or b∈ P. In this paper we discuss the left analogue of these notions and give a relation between completely prime left ideals of a ring R and those of R[x; σ, δ]; where σ is an automorphisms of R and δ a σ-derivation of R. It has been proved that if P is a completely prime left ideal of R such that σ(P) = P and δ(P) ? P, then P[x; σ, δ] is a completely prime left ideal of R[x; σ, δ].